We calculate the all-sky number of galaxy clusters that are expected to be gravitationally lensed by foreground massive clusters .
We describe the redshift and number distributions of clusters using a Press-Schechter analysis , and model the foreground lensing clusters as singular isothermal spheres .
If \Omega _ { m } = 0.3 and \Omega _ { \Lambda } = 0.7 , we expect \sim 30 cluster-cluster strong lensing events that involve foreground X-ray luminous clusters with total mass greater than 7.5 \times 10 ^ { 14 } h ^ { -1 } { M _ { \hbox { $ \odot$ } } } , or X-ray luminosity L _ { x } ( 2 - 10 { keV } ) \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { % $ \sim$ } } } \hbox { $ > $ } } } 8 \times 10 ^ { 44 } h ^ { -2 } { ergs s ^ { -1 } } , and background clusters with total mass greater than 10 ^ { 14 } h ^ { -1 } { M _ { \hbox { $ \odot$ } } } .
The number expected in an open universe with \Omega _ { m } = 0.3 is less than \sim 4 .
Because of uncertainty in \sigma _ { 8 } , the root-mean-square density fluctuations in spheres of radius 8 h ^ { -1 } Mpc , the exact number of such lensing events is uncertain by a factor of about 5 .
We examine methods to detect cluster-cluster lensing events based on optical , X-ray , and Sunyaev-Zel ’ dovich effect observations .